Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither.
$$\mathbf { u } = \langle 5,6 , - 8 \rangle , \mathbf { v } = \langle 10,12 , - 16 \rangle$$
A) parallel
B) orthogonal
C) neither

Question

Quizplus · Accepted Answer

Two vectors are parallel if one is a scalar multiple of the other. In other words, if we can write one vector as a constant multiple of the other, they are parallel.

To check if $\mathbf { u }$ and $\mathbf { v }$ are parallel, we can check if there exists a scalar $c$ such that $\mathbf { u } = c\mathbf { v }$ or $\mathbf { v } = c\mathbf { u }$.

We can check if $\mathbf { u }$ is a scalar multiple of $\mathbf { v }$:
$$\begin{aligned} \mathbf { u } &= c \mathbf { v } \ \langle 5,6 , - 8 \rangle &= c \langle 10,12 , - 16 \rangle \ \end{aligned}$$
The system of equations that gives us the value of $c$ is:
$$\begin{cases} 5 = 10 c \ 6 = 12 c \ -8 = -16 c \end{cases}$$

Solving for $c$, we get:
$$ c = \frac{1}{2} $$
Since we found a value of $c$ such that $\mathbf { u } = \frac{1}{2}\mathbf { v }$, we conclude that $\mathbf { u }$ and $\mathbf { v }$ are parallel.

Determine Whether u\mathbf { u }u And v\mathbf { v }v Are Parallel, Orthogonal, or Neither

Determine Whether $\mathbf { u }$ And $\mathbf { v }$ Are Parallel, Orthogonal, or Neither